REGENERATION THEORY 



145 



For, from the eciuation at top of page 298 '■'' 

 w(zo - As) - 7t<So) 1 C AJ{i\) 



Az 



^ Az 



J_ r AJ(i\)d{i\) 



2iri J, (i\ — 2o — Az){i\ — ZqY 



It is required to show that the integral exists. Now 



r AJ{i\)d{i\) 



J J (/X — 2o — A2)(iX — zo)^ 



AJ{i\)d{i\) 



A-o > 0. (64) 



-J 



(?X - 20)=* 



1 + 



i\ — Z( 



+ 



A22 



■i\ 



Zo 



+ etc. 



(65) 



if A2 is taken small enough so the series converges. It will be sufficient 

 to confine attention to the first term. Divide the path of integration 

 into three parts, 



00 <X< — |2o| — 1, 



-1<X< 



+ 1, 



+ 1 <X<oo 



In the middle part the integral exists because both the integrand and 

 the range of integration are finite. In the other ranges the integral 

 exists if the integrand falls off sufficiently rapidly with increasing X. 

 It is sufficient for this purpose that condition (BI) be satisfied. The 

 same proof applies to 77(2). 



Next, consider lim ^(2) = iv(iy). If iy is a point where J{iy) is 



continuous, a straightforward calculation yields 



w{iy) = AJ{iy)l2 + P{iy). (66a) 



Likewise, 



w(nO = - AJ{:iy)l2 + P{iy) (66b) 



where P{iy) is the principal value ^* of the integral 



1 r^,^i,^. 



Zirt J iX — ty 

 Subtracting 



zv(iy) — n{iy) = AJ{iy) 



If (iy) is a point of discontinuity of J{iy) 



\w\ and 1 77 1 increase indefinitely as X 

 Next, evaluate the integral 



(67) 

 (68) 



J'iv(z)e'''dz, 

 X+I 



" E. W. Hobson, " Functions of a Real Variable," vol. I, 3d edition, § 352. 



